Summary: Modern-day techniques for designing neural network architectures are highly reliant on trial and error, heuristics, and so-called best practices, without much rigorous justification. After choosing a network architecture, an energy function (or loss) is minimized, choosing from a wide variety of optimization and regularization methods. Given the ad-hoc nature of network architecture design, it would be useful if the optimization led to a sparse solution so that one could ascertain the importance or unimportance of various parts of the network architecture. Of course, historically, sparsity has always been a useful notion for inverse problems where researchers often prefer the \(L_1\) norm over \(L_2\). Similarly for control, one often includes the control variables in the objective function in order to minimize their efforts. Motivated by the design and training of neural networks, we propose a novel column space search approach that emphasizes the data over the model, as well as a novel iterative Levenberg-Marquardt algorithm that smoothly converges to a regularized SVD as opposed to the abrupt truncation inherent to PCA. In the case of our iterative Levenberg-Marquardt algorithm, it suffices to consider only the linearized subproblem in order to verify our claims. However, the claims we make about our novel column space search approach require examining the impact of the solution method for the linearized subproblem on the fully nonlinear original problem; thus, we consider a complex real-world inverse problem (determining facial expressions from RGB images).